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Chapter 4: Problem 82

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\frac{1}{2} \tan (\pi x+1)$$

Short Answer

Expert verified
The function \(y = \frac{1}{2}\tan(\pi x + 1)\) plots a wave that completes one full cycle in the interval of one unit on the x-axis. To show two periods, choose a viewing window that displays the graph from -1 to 1 on the x-axis and an appropriate interval on the y-axis that captures all highs and lows of the function within this interval.

Step by step solution

01

Identify the period

First, let's identify the period of the function. The standard period of the tangent function, \(\tan(x)\), is \(\pi\). However, the inside of the function is multiplied by \(\pi\), which means the actual period of our given function will be 1. That is, the function completes one cycle in the interval of one unit.
02

Determine the viewing window

The next step is to determine the viewing window. A full period of the function extends from -0.5 to 0.5. Since we need to show at least two periods of the function, we'll extend the window to display the function from -1 to 1 on the x-axis. For the y-axis, select an interval that displays all the highs and lows of the function within the chosen x-axis interval.
03

Plot the function

Finally, we can plot the function using a graphing utility. The values of \(y = \frac{1}{2}\tan(\pi x + 1)\) have to be calculated for selected values of x between -1 and 1. The resulting points can be plotted and connected with smooth, continuous curves. Note that the function will have vertical asymptotes at the end points of each period.

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Most popular questions from this chapter

In Chapter \(5,\) we will prove the following identities: $$ \begin{aligned} \sin ^{2} x &=\frac{1}{2}-\frac{1}{2} \cos 2 x \\ \cos ^{2} x &=\frac{1}{2}+\frac{1}{2} \cos 2 x \end{aligned} $$ Use these identities to solve. Use the identity for \(\sin ^{2} x\) to graph one period of \(y=\sin ^{2} x\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a tangent function to model the average monthly temperature of New York City, where \(x=1\) represents January, \(x=2\) represents February, and so on.

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=-4 \cos \left(2 x-\frac{\pi}{2}\right)$$

Use a vertical shift to graph one period of the function. $$y=2 \sin \frac{1}{2} x+1$$

What does a phase shift indicate about the graph of a sine function? How do you determine the phase shift from the function's equation?
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